Week 2 Numerical Descriptive Measures int

Statistic for Business

Week 2

Agenda

Time Activity

90 minutes Central Tendency 60 minutes Variation and Shape

Objectives

By the end of this class, student should be able to understand:

• How to measures central tendency in statistics

Numerical Descriptive Measures

Central Tendency

Variation and Shape Measures for

a Population

The Covariance

and The Coefficient of

Central Tendency

Mean (Arithmetic

Mean)

Median

Mean

Consider this height data:

160 157 162 170 168 174 156 173 157

Mean

Sample size

n

Observed values The ith _value

Mean

How about this data of business statistic’s

students monthly spending:

What is the MEAN?

Monthly Spending Frequency

less than Rp. 500.000 2

Mean

In this case we can only ESTIMATE the MEAN…

Keyword: “MIDPOINTS”

Spending Frequency

less than Rp. 500.000 2

Estimated Mean

Midpoint Frequency Mid * f

250000 2 500000

750000 7 5250000

1250000 13 16250000

1750000 5 8750000

Total 27 30750000

Mean

The following is “Student A” Score:

What is the average score of “Student A”?

Course Credits Score

Business Math 3 60

English 2 80

Organization Behavior 3 100

Statistics 4 90

Mean

Consider these two sets of data:

A 150 152 154 155 155

155 155 155 155 157 Mean?

B 150 152 154 155 155

It is DANGEROUS to ONLY use

MEAN in

Median

Consider these two sets of data:

A 150 152 154 155 155

155 155 155 155 157 Median?

B 150 152 154 155 155

Median

What is the median of this height data:

160 157 162 170 168 174 156 173 157

How about this data:

Median

How about this data of business statistic’s

students monthly spending:

What is the MEDIAN?

Monthly Spending Frequency

less than Rp. 500.000 2

Median

The MEDIAN group of monthly spending is Rp.

1.000.000 but less than Rp. 1.500.000

Or

ESTIMATE the

Estimated Median

Estimated Median = Rp. 1.173.076,92

Monthly Spending Frequency

less than Rp. 500.000 2

Mode

What is the mode of this height data:

160 157 162 170 168 174 156 173 157

How about this data:

Mode

How about this data of business statistic’s

students monthly spending:

What is the MODE?

Spending Frequency

less than Rp. 500.000 2

Mode

The MODAL group of monthly spending is Rp.

1.000.000 but less than Rp. 1.500.000

But the

actual Mode may not even be in that

Mode

Without the raw data we don't really know…

Estimated Mode

Estimated Mode = Rp. 1.214.285,72

Spending Frequency

less than Rp. 500.000 2

Central Tendency

Central Tendency

Arithmetic

Middle value in the ordered array

Most

3.10

This is the data of the amount that sample of nine customers spent for lunch ($) at a fast-food restaurant:

4.20 5.03 5.86 6.45 7.38 7.54 8.46 8.47 9.87

3.12

The following data is the overall miles per gallon (MPG) of 2010 small SUVs:

24 23 22 21 22 22 18 18 26

26 26 19 19 19 21 21 21 21

21 18 29 21 22 22 16 16

Compounding Data

Interest

Rate

Growth

Rate

Compounding Data

Suppose you have invested your savings in the stock market for five years. If your returns each year were 90%, 10%, 20%, 30% and -90%, what would your average return be during this

Compounding Data

If we use arithmetic mean in this case

Geometric Mean

This is called geometric mean rate of return

1

Measure of Central Tendency For The Rate Of Change Of A Variable Over Time:

The Geometric Mean & The Geometric Rate of Return

 Geometric mean

 Used to measure the rate of change of a variable over time

 Geometric mean rate of return

 Measures the status of an investment over time

Geometric Mean

Lets reconsider the previous problem. We knew that we invest $100 in year 0 (zero). However, by the end of year 5 the value of the stock became $32.6. Calculate the annual average return!

•

$100

Year 0

•

$32.6

Geometric Mean

Population of West Java

Population of West Java:

• Year 2000: 35.729.537

• Year 2010: 43.053.732

3.22

3.22

a. Compute the geometric mean rate of return per year for platinum, gold, and silver from 2006 through 2009.

Variation and Shape

Range

Variance and Standard Deviation

Coefficient of Variation

Z Scores

Review on Central Tendency

Consider this data:

160 157 162 170 168 174 156 173 157 150

Range

Consider this data:

160 157 162 170 168 174 156 173 157 150

Range

min

max

X

X

Measures of Variation:

Why The Range Can Be Misleading

 Ignores the way in which data are distributed

 Sensitive to outliers

Variance and Standard Deviation

Variance

Standard

Deviation

Let’s see this data again:

160 157 162 170 168 174 156 173 157 150

What is the mean?

Variance and Standard Deviation

Data Deviation (Dev)^2

160 -2.7 7.29

Variance and Standard Deviation

• Sample

• Population

Measures of Variation:

Comparing Standard Deviations

Standard Deviation

How about this data of business statistic’s

students monthly spending:

What is the STANDARD DEVIATION?

Monthly Spending Frequency

less than Rp. 500.000 2

Standard Deviation

How about this data of business statistic’s

students monthly spending:

What is the STANDARD DEVIATION?

Monthly Spending Frequency

less than Rp. 500.000 2

Rp. 500.000 but less than Rp. 1.000.000 7 Rp. 1.000.000 but less than Rp. 1.500.000 13 Rp. 1.500.000 but less than Rp. 2.000.000 5

Estimated Standard Deviation

Midpoint Frequency Dev^2 (Dev^2)*f

250000 2 790123456790.12 1580246913580.25

750000 7 151234567901.24 1058641975308.64

1250000 13 12345679012.35 160493827160.49

1750000 5 373456790123.46 1867283950617.28

Total 27 4666666666666.67

𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 = 4666666666666.67 ₂₇ = 172839506172.84

The Coefficient of Variation

The Coefficient of Variation

Let’s see this height data again:

160 157 162 170 168 174 156 173 157 150

What is the mean and standard deviation

The Coefficient of Variation

Students with height before is weighted as follows:

50 55 57 52 55

69 60 65 71 70

What is mean and standard deviation?

The Coefficient of Variation

Height Weight

Mean 162.7 60.4

SD 8.125 7.8

Which one has more variability?

Coefficient of Variation: CV_Height = 4.99%

Locating Extreme Outliers: Z Score

Let’s see this height data again:

160 157 162 170 168 174 156 173 157 150

160 162 164 150 152 154 156 158

Mean = 162.7

166 168 170 172 174

Locating Extreme Outliers: Z Score

Therefore, Z Score for 160 is?

160 162 164 150 152 154 156 158

Mean = 162.7

166 168 170 172 174

Z Score_162.7 = 0

Z Score_154.6 = -1 _{Z Score}

170.8 = 1

Locating Extreme Outliers: Z Scores

Let’s see this height data again:

160 157 162 170 168 174 156 173 157 150

What is the Z Scores of 160, 174, 168 and 150?

Locating Extreme Outliers: Z Score

SD

X

X

Z

_X





• A data value is considered an extreme outlier if its Z-score is less than -3.0 or greater than +3.0.

Shape

Let’s see this height data again:

160 157 162 170 168 174 156 173 157 150

160 162 164 150 152 154 156 158

Mean = 162.7

166 168 170 172 174

Shape

What if the height data is like this:

163 168 162 170 168 174 156 173 157 150

160 162 164 150 152 154 156 158

Mean = 164.1

166 168 170 172 174

Median = 165.5

Shape

Describes how data are distributed

Mean = Median

Mean < Median Median < Mean

Exploring Numerical Data

Quartiles

Interquartile Range

Quartiles

Let’s consider this height data:

160 157 162 170 168 174 156 What is the Q₁, Q₂ and Q₃?

Q₁ = 157

Quartiles

Let’s then consider this height data:

160 157 162 170 168 174 156 173 150 What is the Q₁, Q₂ and Q₃?

Q₁ = 156.5

Quartiles

And this height data:

160 157 162 170 168 174 156 173 157 150 What is the Q₁, Q₂ and Q₃?

Q₁ = 157

Quartiles

160 162 164 150 152 154 156 158

Median = 161

166 168 170 172 174

Q₁ = 157 _Q

3 = 170

25% 25% 25% 25%

Interquartile Range

160 162 164

150 152 154 156 158 166 168 170 172 174

Q₁ = 157 _Q

3 = 170

50%

Interquartile Range

160 162 164

150 152 154 156 158 166 168 170 172 174

Q₁ = 157 _Q

3 = 170

What is the Interquatile range?

Interquartile Range

1 3

Q

Q

Range

ile

Five-Number Summary

max 3

1

min

Q

Median

Q

X

Five-Number Summary

Let’s see again this height data:

160 157 162 170 168 174 156 173 157 150 What is the Five-Number Summary?

Boxplot

Boxplot

Let’s see again this height data:

160 157 162 170 168 174 156 173 157 150

Boxplot for the Height of Business

Statistic’s Student 2014

150 157 161 170 174

Distribution Shape and The Boxplot

Right-Skewed

Left-Skewed Symmetric

Karl Pearson's Measure of Skewness

S

Median X

Bowley's Formula for Measuring

Skewness

150 157 161 170 174

3.10

This is the data of the amount that sample of nine customers spent for lunch ($) at a fast-food restaurant:

4.20 5.03 5.86 6.45 7.38 7.54 8.46 8.47 9.87 a. Compute the mean and median.

b. Compute the variance, standard deviation, and range c. Are the data skewed? If so, how?

3.62

3.62

73 19 16 64 28 28 31 90 60 56 31 56 22 18 a. Compute the mean, median, first quartile, and third quartile.

b. Compute the range, interquartile range, variance, and standard deviation.

c. Are the data skewed? If so, how?

3.22

3.22

a. Compute the geometric mean rate of return per year for platinum, gold, and silver from 2006 through 2009.

3.66

3.66

a. For each variable, compute the mean, median, first quartile, and third quartile.

b. For each variable, compute the range, variance, and standard deviation

c. Are the data skewed? If so, how?

d. Compute the coefficient of correlation between calories and total fat.